On Fri, 8 Nov 2013, William Elliot wrote:> On Sat, 9 Nov 2013, Victor Porton wrote:> > > >> > Are these correct?> > > Definition 3.60.> > > > > A thorning of a point a in a complete lattice L> > > is some A subset L\bottom for which sup A = a> > > and for all x,y in A, bottom /= x inf y.> > > (bottom is minimum element of L; S\a used for S\{a})> > > > > > {a} is a thorning of a.> > > > Definition 3.61.> > > > A weak partition of a point a in a complete lattice L> > > is some A subset L\bottom for which sup A = a> > > and for all x in A, bottom /= x inf (sup A\x).> > > > > > {a} is not a weak partition of a.> > > > > >> > Definition 3.62.> > > > Correct.> > > > > A strong partition of a point a in a complete lattice L> > > is some A subset L\bottom for which sup A = a and> > > for all U,V subset S, (U,V disjoint iff bottom /= (sup U) inf (sup V)).> > > > > > There are no strong partitions because if U = A and V is empty> > > bottom = (sup U) inf (sup V).> > > > U and V must be non-empty.> > > You need to state that in the definition.> > A strong partition of a point a in a complete lattice L is> some A subset L\bottom for which sup A = a and for all > not empty U,V subset S, (U,V disjoint iff bottom /= (sup U) inf (sup V)).> > > Every weak partition is a strong partition. Consequently strong partitions > > exist.> > According to Obvious 3.63> Strong partitions are weak partitions> and weak partitions are thorning.> > Thus by what you say, strong and weak partitions are the same.> > What's obvious to me is that strong partitions are thornings. > Do you have a proof that weak partitions are thornings or strong partitions?> Strong partitions are not weak partitions for {a} (as above) is > a strong partition of a. It is not a weak partition of a.

Here's an example of a weak portition that's not a thorning.Let L = [0,2]^2 with coordinate wise ordering.A = { (x, 2-x) | x in [0,2] } is a weak partiton of (2,2)because for all p in A, sup A\p = (2,2).

It's not a thorning because (0,2) and (2,0) in A.For the same reason, it's not a strong partition.

> > > Where in the heck is Conjecture 4.153? In what section?> > > > I don't understand your order.> > Where, in your text, do I find Conjecture 4.153, that a filter > can be partitioned into ultrafilters in the REVERSE order.> > > > There is no thorning of a filter by ultrafilters.> > > The set of all ultrafilters below a filter (in the reverse order) is a > > thorning of this filter.> > If G,H are distinct ultrafilters for S, > filter F subset G,H, then G sup H = P(S).> > Reversing that, for all distinct ultrafilters G,H <= F, G inf H = 0.> Thus a thorning of F by ultrafilters can have only one element > and there are no thornings for filters that aren't ultra.> > > > Why the names thorning and partition for> > > defintions that are unrelated to the words.> > > > "Thorn" means to roughly thorn without proper "boundaries" unlike > > partitions.> > Makes no sense. Partitions are dividing into parts and there's no> sense of that in your definitions.> > A thorn is pointed part of a plant designed to scrach or prick.> For example, blackberry thorns, rose bush thorns.> There is no verb thorning. Thorns and the definition > of thorning are grossly mismatched.> >